Sums of Squares of Integers is the first book to combine elementary methods with analytic methods of modular functions to study the representation of integers as sums of squares. It is the first book to tell how to compute the number of representations of an integer n as the sum of s squares for any s and n. This book is also the first to give a proof of Szemeredi's theorem, an important result in modern number theory. It discusses how the modern theory of modular forms complements and clarifies the classical, fundamental results about sums of squares. This text includes applications to such diverse fields as cryptography, crystallography, and thermal radiation. Written by world-class experts, this book is ideal for mathematicians, particularly number theorists and computer scientists.